I've finally knitted enough so that I can now transfer the work onto an extra piece of yarn, from the needles, so I can knit the body and sleeves separately.

Thing is, I'm not sure how to go about doing that without having everything fall apart. Should I slide the needles out and just be really careful? Should I do it stitch by stitch? Should I get a smaller tapestry needle and thread it all through, THEN remove the needles?

You can slide it through your stitches while they're on the needle. Just put the waste yarn on a tapestry needle and go through the stitches in the little space under the needle. You can do several at a time and it goes fast. Once the yarn is through all the stitches, take out the needle.

Okay, I'm getting really really paranoid about it. I tried it with a few stitches, and it just made it all look tattered or something. And I'm always worried I'm going to hook the tapestry needle into a loop, instead of just in that space.

Do you think it would work to just slip the stitches straight onto needles? I'm supposed to try it on, but I'm positive it fits. I could just put a needle through each of the sleeves, and the body, and work them that way ... right? (Also, I don't do anything to the front, do I?) pattern

From looking at the pattern, you should still slip your sleeve stitches to yarn since you work the body parts together first. Otherwise, you can just slip all the body stitches on your needles, do is as if to purl so you don't twist them.

For the sleeve stitches, use your tapestry needle as if it is a knitting needle and slip the stitches onto it and let the loops work their way down onto the yarn.

And yes, you put the fronts on the needle, too. At this point in the pattern, you're connecting the front and back and basically knitting down from the armpits to the hem. The sleeve stitches will be worked later. It would be difficult to work the body with the sleeves on needles since they'd get in the way. If you slip the sleeve stitches as I said above, you shouldn't have a problem. (When they're on the yarn, they look like they tighten up, but when you put the needle back in, you'll be ok.)